# eigenvectors of the square of an nxn matrix

I am aware of the relationship between the eigenvalues of a matrix and the matrix square (being the eigenvalues squared) but what relationship, if any, is there between the eigenvectors of the matrix and its square? Does it depend on what type of matrix i.e. hermitian, unitary etc?

• Any eigenvector of $A$ is an eigenvector of $A^2$. – Lord Shark the Unknown May 26 '17 at 17:05

If A is a linear transformation from vector space V to itself, with eigenvalue $\lambda$, there exist a non-trivial vector v, such that $Av= \lambda v$. Then $A(Av)= A(\lambda v)$. So $A^2v= \lambda (Av)= \lambda(\lambda v)= \lambda^2 v$. That is if $\lambda$ is an eigenvalue of A then $\lambda^2$ is an eigenvalue of $A^2$.
• It's worth noting that an eigenvector of $A^2$ doesn't need to be an eigenvector of $A$, keeping in mind nilpotent matrices and generalized eigenvectors. – Roland May 26 '17 at 17:23